Conditions for universal reducibility of a two-stage extremization problem to a one-stage problem Dedicated to the memory of Richard Bellman An extremization problem in the general form maxxX f (x) is consid ered. This notation is treated as the description of a parametric family of problems where the extremized function f is the same but the set X is a variable parameter. Such a “mass” treatment of the extremization problem for function f determines implicitly the “choice transformations” X Y, where Y is the set of solutions, i.e., Y = Arg maxxX f (x), and X goes over a given family X of admissible sets. Similarly, a two-stage problem of sequential extremization of two functions, and, determines the su perposition of two related choice transformations. We consider the cases when the function and/or may be vectorial and the extremization is understood in the Pareto sense. The very possibility of reducing a two-stage problem to a one-stage problem having the same solution set Y for every admissible X X is studied. Unlike the usual “lexicographical” extremiza tion of two scalar functions, such a reduction is not always possible in the vectorial case. The necessary and sucient conditions for it are stated.

max f (x). (1) xX In this way we shall treat the formulation (1) in two ways: (1) as the individual problem of extremization of the given function f on the given set X, and (2) as the mass problem of extremization of the same function f on various sets X in which role the members of some given family X of admissible sets are taken. In what follows, under the problem of extremization for function f (briey, f -problem), we shall understand the mass problem of extremization for f in the above-mentioned sense. That is, we shall virtually consider the parameterized (by X) family of individual problems in the form (1) which are dierent in the parameter X X.

We shall be interested in the behavior of the set Y of solutions of the f -problem (1) under all possible X’s, i.e., in the form of transformation X Y implicitly dened due to (1) where X X and Y = Arg max f (x). (2) xX Indeed, similar transformations converting every admissible set X into some subset Y X really appear as a result of applying certain “mass” procedures to the sets X X (such procedures may specically include the use of extremizing operations for various functions). Every problem (procedure) which explicitly or implicitly selects from any set X X some subset Y is called a choice problem on X X. In these terms the simplest f -problem of extremization (1) may be considered as a paradigm of choice problems. The aim of the present work is to consider some other problems having more complicated forms and to collate them with this paradigm, being interested in the possibility or impossibility of nding the “equivalent” paradigm problem for any such starting problem (the sense of “equivalence” being strictly dened below).

As the starting problems we shall furthermore consider the problems of sequential extremization of two functions, and, having the same domain U. We shall assume that under any xed X (X U ) used as the admissible (feasible) set for the rst stage of extremization, viz., the -problem, the received solution set V for -problem is used further as the admissible set for the second stage of extremization, viz., the -problem.

The resulting overall problem max (v), where V = Arg max (x), (3) vV xX Conditions for universal reducibility will be called the two-stage problem of sequential extremization of functions and (or the, -problem for brevity). Such a, -problem generates the choice transformation X Y, X X, where Y = Arg max (v). (4) vArg maxxX (x) It is evident that this transformation is the superposition of two “single stage” choice transformations, X V and V Y, generated by the -problem and the -problem, respectively.

Let us pose the question: is it possible to replace a two-stage, problem (3) by some equivalent one-stage f -problem? The answer to this question essentially depends on the way of dening the notion of “equiv alent substitution”. It is natural to require that the “substitute” problem possess exactly the same set of solutions as the initial one. But if we applied this requirement only to the individual problem on a single xed set X and if we simultaneously permitted arbitrary selection of any criterial function f in the proposed substitute equivalent f -problem, then the answer to our question should turn out to be trivially positive. It would be sucient, e.g., to take the characteristic function of the set Y as the f function, i.e., let f (x) = 1 for x Y and f (x) = 0 otherwise. Nevertheless the above question remains nontrivial if we treat our problems as mass problems of choice;

then according to such a treatment we shall consider all admissi ble sets X X, and by the “equivalence of problems” we shall mean the coincidence of their respective solution sets, Y ’s, for every X X.

Denition 1. Consider two choice problems on sets X from the same family X. We shall say that one problem is universally reducible to another (and conversely) if for every X X the respective solution sets Y1 and Y of these two problems coincide.

In what follows, speaking of reducibility of one problem to another (specically, of two-stage to one-stage), we shall mean the universal redu cibility in Denition 1 even if the word “universal” is omitted.

Remark. Although, as we have indicated above, for every choice prob lem on a xed X a respective f -problem with the same solution set Y can be designed in a trivial way, such a design by itself yields nothing for the answer to the question on the universal reducibility to an f -problem.

Indeed, the so designed critical function f will generally depend para metrically on X, i.e., f = fX (x). The question is if it is possible to “sew” dierent functions fX (x) together to get a joint function f (x) on XX X independent of X as is required in the form (1) of the mass f -problem.

In the choice theory (see, e.g., [87]) for this question in a general case the negative answer is given: denitely not every choice problem is universally reducible to a scalar or even vectorial extremization problem. The result 338 III. Теория принятия решений of analysis of possibilities for such reducibility for two-stage, -problems are given below (some of them have been published earlier, particularly in [84, 87]).

In the simplest case, when both and are scalar functions, the two stage, -problem (3) is a well-known problem of “lexicographic maximiza tion” (see, e.g., [134]). Such a “scalar-scalar” two-stage problem is always reducible to a one-stage extremization problem (1) with some function f being a scalar one also;

we shall discuss it below. But if in the role of and/or, one may use vectorial functions then the situation becomes more complicated: universal reducibility of a, -problem (3) to an f -problem (1) can demand that certain conditions be fullled. The setting of such conditions is the main topic of this paper.

Everywhere below we assume for the sake of simplicity that all consid ered functions (f,, etc.) are dened on some nite set U (i.e., f : U R or : U Rn, etc.) and the family X is the collection of all nonempty subsets X of the set U (i.e., X = 2U \{ }).

2. Vectorial extremization Let us start from statements of extremization problems under several criterial functions. Assume that n scalar functions 1,..., n on U are given. Before looking at the “sequential” application of dierent criterial functions in extremization procedures, let us discuss their “parallel” appli cation. The “parallel” (i.e., simultaneous and “equitable”) usage of functions 1,..., n as extremization criteria naturally leads to the usual considera tion of the “vectorial” criterial function = (1,..., n ) when the vector maximization problem max (x) (5) xX is understood as the problem of nding the solution set Y = Arg max (x) = xX (6) = {y X there is no x X such that (x) (y)}.

Here vectorial inequality for is dened as an appropriate generalization of scalar inequality which converts itself into usual numerical inequality in the case n = 1. This fact implies that the scalar extremization problem is a particular case of the vectorial one. Following the standard denition we shall treat vectorial inequality as a component-wise problem (a vectorial superiority relation in the Pareto sense):

(x) (y) (i (x) i (y), i = 1,..., n). (7) So Y in (6) is the set of Pareto-maximal points for the vectorial function.

Conditions for universal reducibility Remark. The denition of the Paretian relation of strict vectorial supe riority usually admits that some (but not all) respective component-wise inequalities may be unstrict. For further exposition this dierence in two varieties of vectorial inequality is unessential.

Let us pose the question: is the vectorial maximization problem (5), with n 1, reducible to some scalar maximization problem, with n = 1?

The answer to this question will serve as an illustration for the notion of universal reducibility for mass choice problem.

First, we shall introduce some notations. The failure (logical negation) of the vectorial inequality we shall call vectorial nonsuperiority and denote by (in the scalar case, n = 1, the relation is equivalent to ). Let us also introduce for vectors the following relation of mutual nonsuperiority denoted by :

Let us satisfy ourselves that under the existence of a -triad in U the universal reducibility of the -problem (5) to an f -problem (1) with some scalar function f is denitely impossible. Really, on setting X = {u, v, w}, we get for (5) Y = {v, w}. Hence in the case of the reducibility of (5) to (1), we might have f (u) f (v) = f (w). On the other hand, on setting X = {u, w}, we get for (5) Y = {u, w}. In the case of the reducibility of (5) to (1) this would imply that f (u) = f (w), in contradiction with the preliminary result f (u) f (w). Therefore the problem (5) is not reducible to (1).

Remark. Really, the converse is also true: the existence of a -triad in U is not only sucient but also necessary for the irreducibility of a problem (5) to (1).

This well-known fact is easily deduced in Section 5 by using some elementary notions of decision theory 1. In the main part of the text we shall if possible avoid addressing these notions for the sake of autonomy of exposition. For the same purpose in several cases we give independent proofs of statements known (in one form or other) in the theory of choice and decision making [84, 87, 134, 221].

1 In this case the essence of the matter is the nontransitivity of the mutual nonsupe riority relation which is just equivalent to the existence of a -triad in U.

340 III. Теория принятия решений In particular, we shall now give an independent “constructive” proof of the fact that if -triads in U are absent then the -problem (5) is reducible to an f -problem (1) with some scalar function f. It will be useful for us to write the requirement of absence of -triads in U in the form of the Quadruple condition: for every x, y, r, s U (y) (s) (r) (s).

(x) (y), (x) (r), Really, the existence of a -triad u, v, w U of the form (8) obviously violates the Quadruple condition (it is sucient to take x = r = v, y = u, s = w). Conversely, let the Quadruple condition be violated, viz., its left part be fullled but its right part be violated (i.e., (r) (s)). Then, as it is easy to see, under any possible relation between (x) and (s) in the quadruple x, y, r, s at least one -triad (x, y, s or x, s, r) will exist.

Thus let -triads be absent in U so that the Quadruple condition is fullled. We shall design the desired function f on U. Dene for every u U the set E(u) = {v U (u) (v)} and let n f (u) = i (v).

|E(u)| vE(u) i= We will show that the given scalar function f is equivalent to the vectorial function in the sense that (x) (y) f (x) f (y).

Note for the beginning that due to the absence of -triads in U (x) (y) E(x) = E(y), hence (x) (y) f (x) = f (y).

On the other hand, the denition of f in light of the Quadruple condi tion immediately implies (x) (y) f (x) f (y).

This together with the previous implication just means the equivalence between f and which yields universal reducibility of the -problem to the f -problem.

Denition 2. A vectorial extremization problem (5) (with n 1) will be called essentially vectorial if it is not universally reducible to an f -problem (1) with any scalar function f.

Conditions for universal reducibility Now the above statement on reducibility of problem (5) to (1) may be summarized in the following Lemma.

Lemma 1. For a problem of extremization of a vectorial function on U (with X = 2U \{ }) to be essentially vectorial, it is necessary and sucient that -triads were in U.

Remark. We once more emphasize that nonreducibility of a given vecto rial extremization problem to a scalar one is generated by the requirement of universality of such a reducibility on all X X. In contradistinction to it, the statements on the reducibility of a vectorial extremization problem to a scalar one on a single xed set X are sometimes considered. These statements can be nontrivial if we set some a priori requirements on the form of the scalar function sought (e.g., the representability in the form of linear combination of components of the initial vectorial function).

3. Sequential extremization: the simple cases Now let various criterial functions, scalar or vectorial, be used in ex tremization procedures sequentially. We shall consider the two-stage pro cedure in the form of a, -problem (3) with criterial functions and at the rst and the second stages of extremization, respectively. The four types of problems depend on the types of functions and :

I. Let us start with a scalar-scalar problem (type I, “lexicographic scalar extremization”) [221]. We shall display its universal reducibility 342 III. Теория принятия решений to a scalar extremization problem by using the following self-explanatory construction. Let |U | = N. Then without loss of generality we may consider values and to be integers between 0 and N 1 and index them by N -ary digits. (Here we as usual base our results on the evident fact that functions under extremization admit any monotone transformations without changing the solutions for extremization problems).

Consider now two-digit numbers in the N -ary positional calculus sys tem having the form,, i.e., where digit (one-digit number) is set in a higher position and in a lower one. Let us construct now a new scalar function on the set U having values (x) = (x), (x), x U, (9) and consider the extremization problem max (x). (10) xX It is easy to make sure that the, -problem (3) with given scalar functions and is universally reduced to the -problem (10). Indeed, the set Y = Arg maxxX (x) obviously consists of those and only those elements y X which satisfy the two conditions:

(1) among elements of X, they have the maximal value of the highest digit for, i.e., the maximal value of ;

(2) among these -maximal elements of X, they have the maximal value of the lowest digit for, i.e., the maximal value of.

But this is exactly the description of the set Y given by (4).

II. Now, let in a two-stage, -problem (3), the function be a scalar and be a vectorial one: = (1,..., n ). Repeating the process of introducing two-digit N -ary numbers, we shall dene a vectorial function = (1,..., n ) by the equalities i (x) = (x), i (x), i = 1,..., n. (11) Again it is possible to verify that a two-stage, -problem (3) is reduced to the one-stage -problem (10) but now with a vectorial function of the form (11). For this purpose we shall consider an arbitrary element y X and show that it is not a solution of the problem (10) if and only if it is not a solution of the problem (3).

Denote = maxxX (x). Then the set V dened by (3) is V = {v X|(v) = }. For y X not to be a solution of the -problem (10), it is necessary and sucient that for some z X, the inequality (y) (z) holds, i.e., ( (y), 1 (y),..., (y), n (y) ) ( (z), 1 (z),..., (z), n (z) ), (12) Conditions for universal reducibility which is possible in exactly one of the two cases:

(y) ;

in this case (12) holds with any z V ;

(a) (y) =, but there exists z V such that (y) (z).

(b) Obviously, (a) is equivalent to the fact that in the, -problem (3), y has not been included in the set of solutions already at the rst stage, and (b) is equivalent to the fact that y is included in the set of solutions at the rst but not at the second stage of (3). So (a) and (b) together exhaust the cases when y is not a solution of (3).

The results of Subsections I and II can be summed up in the following theorem.

In concluding this section we shall note that under reduction of a two-stage scalar-vectorial extremization problem to a one-stage problem, the case is possible when the resulting one-stage problem turns out to be not a vectorial but a scalar one. We shall now give the complete characterization of this case based on Lemma 1 and construction (11) of a vectorial function, to the extremization of which the initial, -problem (3) may be reduced. Due to Lemma 1, a -problem (10) will be essentially vectorial if and only if in U there are no -triads, i.e., no triples u, v, w U such that (u) (v), (u) (w), (v) (w). (13) From the denition of (11) it follows that the last two relations in (13) are equivalent to (u) = (v) = (w), (14) (u) (w), (v) (w), and the rst relation in (13) with rst equality in (14) is equivalent to the vectorial inequality (u) (v). (15) So fulllment of the relation system (14), (15) for some u, v, w U is necessary and sucient for a -problem (10) to be essentially vectorial, i.e., irreducible to a scalar extremization problem. Hence it is also necessary and sucient for an initial, -problem (3) to be irreducible to a scalar problem. Speaking dierently, the reducibility of a, -problem (3) to a scalar one is equivalent to nonexistence of a situation of the form (14), (15) on the set U.

Noting that (15) together with the second line in (14) implies that the triple u, v, w is a -triad, we may give the nal form of the latter statement.

344 III. Теория принятия решений Addendum to Theorem 1. For a two-stage problem of scalar-vectorial extremization to be reducible to a problem of scalar extremization, it is necessary and sucient that no subset of the set U having the same value on its elements (i.e., a subset of the form Uc = {u U |(u) = c}) contains a -triad.

Note that due to the impossibility of -triads for scalar functions, this addendum automatically implies the above-mentioned universal reducibility of any scalar-scalar, -problem to a one-stage problem.

Thus the existence of the “preliminary” scalar extremization stage be fore the vectorial extremization stage in any case does not make the prob lem “essentially more complex” than a single vectorial problem (Theo rem 1). Moreover, introducing the preliminary -stage can even “simplify” the initial vectorial -problem, transforming it into a scalar one (Adden dum to Theorem 1) by “destroying” existing -triads (by virtue of giving unequal -values to their elements).

Now let us go to clarify in what degree an extremization problem is complicated after introducing not the scalar but the vectorial preliminary extremization stage.

4. Two-stage extremization: general case In this section the main statements are given which concern the general case of vectorial-vectorial extremization (problems of type IV), and the case of vectorial-scalar extremization (type III) which turns out to be almost as complex.

Start with a problem of type III. Let = (1,..., n ) be a vectorial and a scalar function on U. One might attempt, by analogy with a problem of type II, to design a “lexicographic” vectorial function with components i = i,, i = 1,..., n. But is easy to verify that the initial two-stage problem (3) denitely is not reducible to the extremization of the function in the nontrivial case when for at least one pair of elements, p, q U, (p) (q) but (p) = (q). (16) Really, in this case for the two-stage, -problem of extremization on X = {p, q} the set Y evidently contains just one of the elements p or q. But for the respective one-stage -problem we have (p) (q) and hence Y = {p, q} on the same X = {p, q}.

Attempts at constructing any other function, scalar or vectorial, to reduce a two-stage, -problem to a one-stage -problem are doomed to failure. Consider the triple u, v, w U such that (u) (v), (u) (w), (v) (w), (17) (u) (w) (v) Conditions for universal reducibility (here the rst line in (17) means that u, v, w is a -triad). Then for the two-stage, -problem on X = {u, v, w} the solution set Y surely includes element w. But on X = {u, w} the respective solution set Y contains the single element u (i.e., Y = {u}) and does not include w. It demonstrates the violation of the property of “rational choice,” the heredity property which may be formulated in the following way: if w X X and w Y where Y is the choice from X, then w Y holds where Y is the choice from X. It is easy to see that the heredity property must be kept on solution sets of any one-stage extremization problem (considered as a mass problem of choice, following the Introduction). Therefore the above-stated, -problem on the set U including the triple u, v, w from (17) denitely cannot be reduced to a one-stage problem of extremization, neither scalar nor vectorial.

Thus all problems of vectorial-scalar extremization and, even more so, all problems of vectorial-vectorial extremization of the general type must be broken up into denitely nonempty classes: problems which do not admit universal reduction to the one-stage problem of scalar or vectorial extremization, and problems which do admit such a reduction. We shall give in this section the formulations of conditions which discriminate these classes of problems. The proofs will be given in the next section.

Remark. In Denition 3 as usual it is not forbidden to take as vectorial functions, their particular case, scalar functions. But we note that if the rst function,, is really scalar, then -triads and, even more so,, -1- -2-, and -3-triads surely do not exist.

We now formulate the main theorem of this work.

Theorem 2. For a vectorial-vectorial extremization, -problem to be universally reducible to a vectorial (or, moreover, to a scalar) one-stage extremization problem, it is necessary and sucient that, -1- and, 2-triads be absent in U (respectively, any, -1-,, -2,, -3, and also, -3-triads were absent in U ).

1 See, e.g., [87]. Such a property also used to be called the Cherno condition or -condition (after Sen [221]).

346 III. Теория принятия решений Remark. Theorem 2 covers the general vectorial-vectorial case of the two-stage extremization problem (type IV). Surely it implies the state ments concerning reducibility of more particular problems, of type I and II, which have been considered above, and also of type III. Being applied to problems of type III, Theorem 2 can be specied in the following way.

In the formulation of Theorem 2,, -inconsistent triads of the 1st, 2nd, and 3rd kind are characterized as -triads satisfying the conditions, re spectively, (1) (u) (w) (v);

(2) (u) = (w) (v);

(19) (3) (u) = (w) = (v).

The additional requirement of the absence of, -3-triads (in the list of conditions for reducibility to a scalar problem) here may be omitted, because due to scalarity of it is fullled automatically. Note that just the -triad with additional condition (19.1) has been considered in this section as a counterexample for the reducibility of a, -problem of type III to the one-stage problem. Furthermore, for problems of type I and II, Theorem immediately gives Theorem 1 and the Addendum to it.

Indeed, under scalarity of no, -inconsistent triads exist, hence reducibility of a problem of type II (in particular, of type I) to a vectorial problem is always guaranteed. Moreover, under scalarity of no, 3-triads exist, hence a problem of type I is always reducible to a scalar problem. It results just in Theorem 1. Finally, reducibility of an arbitrary problem of type II to a scalar problem is equivalent to the absence of, 3-triads (because any, -triads are denitely absent). But relations (14), (15), whose fulllment is excluded by the Addendum to Theorem 1, just compose the denition of a, -3-triad. So the results from Section 3 are essentially included in the generalizing Theorem 2 as particular cases.

5. Proof of the general theorem on the vectorial-vectorial problem The proof given below is founded upon a number of statements con cerning binary relations and kindred notions in the theory of choice and decision making.

A. Representation of binary relations of ordering by nume rical functions. Let on a set U a binary relation be given which v (not u is asymmetrical, i.e., u v);

we shall further call a superiority relation. For the given relation, we construct the associated indierence relation denoted by the symbol and dened by u v (not u v and not u v).

Conditions for universal reducibility A relation on U is called a partial order if it is transitive, i.e., (u v and v w) u w, and a weak order if, moreover, the respective relation is also transitive (here orders are considered in the “strict” version according to the presumed asymmetry of ).

Denition 4. Let a vectorial (possibly scalar) function f and a relation be given. We say that f is consistent with if for every u, v U :

The next two lemmata are as a matter of fact the versions of the well-known Szpilrajn theorem and the Dushnik and Miller theorem [134, 199] (concerning imbedding of partial orders into linear ones ) which are expressed in terms of representing functions.

Lemma 3. For any partial (also including weak) order on U there exists a scalar function f on U consistent with.

Lemma 4. For any weak (or partial) order on U there exists a scalar (resp., vectorial) function f on U which represents.

We shall give here independent “constructive” proofs of Lemmata and 4 using a simple iterative procedure in the spirit of R. Bellman’s dynamic programming, for building up functions f to be found. Let a superiority relation and the associated indierence relation on U be given. Consider the iterative procedure f t+1 (x) = max{f t (x);

max (f t (s) + 1)} (22) sx (we mean that here x U, s U, t = 0, 1,..., and that in the case of the absence of s U such that s x, only the rst term remains in the outer braces in (22)).

Take initial values of f :

f 0 (u) = 0, for all u U. (23) It follows from acyclicity of that in U there exists at least one minimal element, i.e., element u such that for it there are no elements v such that 348 III. Теория принятия решений v. For every minimal element u in U obviously f t (u) 0, for all u t = 0, 1,...

Furthermore, the denition of the procedure (22) in virtue of acyclicity and transitivity of implies the next properties:

Values f t (x) for every xed x do not decrease with t increasing.

(a) Positiveness of the value f t (x) = k for some x and t is equiv (b) alent to existence of a chain x p q... u in U, where u is a minimal element of U and the length of the chain (the number of elements x, p, q,..., u) is equal to k.

for all t (24) (e) It follows from Properties (b) and (c) that for (24) the following estimates are true: T N, and for all x U f (x) N. (25) It follows from Property (d) that the function f on U is consis (f) tent with the relation (it is sucient to consider (22) with (24)).

Property (f) yields the statement of Lemma 3. For Lemma 4 to be proved for the case when is a weak order, it is sucient now to verify that in this case the function f so constructed is not only consistent with but, moreover, does represent. To this end we need to show that if u v then f (u) = f (v). Let u v. Properties of the weak order imply us v, s U.

Finally we shall prove the Lemma for the case when is a partial order. With that end in view we consider the procedure (22) under special initial conditions dierent from (23). Let elements of U be numbered:

Conditions for universal reducibility Applying now the procedure (22) with each of these N sets of initial data, we shall get N function sequences {fit }t=0,1,... (i = 1,..., N ) built by the procedure. Each sequence will possess properties similar to (and partially coincident with) Properties (a)-(f) of the sequence {fit } generated by the procedure (22) under zero initial data (23). For convenience of describing properties of the new sequences {fit } determined from the ith set of initial data (26), we part the set U into two subsets ui or u = ui } and Ui = U \Ui+.

Ui+ = {u U | u Consider the behavior of value sequences {fit (u)}t=0,1,... for u Ui and for u Ui+ separately. It is not dicult to see that two respective subsets {fit (v)}, v Ui, and {fit (w)}, w Ui+, can be constructed by independent application of procedure (22) to subsets Ui and Ui+ separately taken in the role of U in (22) under initial data (23) or (26), respectively. Really, let us consider the cases of Ui and Ui+ taken separately.

(1) Case of Ui. To begin with we note that by the construction of sets Ui+ and Ui, due to the transitivity of, the following holds:

If v Ui, and w Ui+, then it is impossible that v w. (27) Hence the procedure (22) being applied to the total set U under initial data (26) works on the subset Ui independently of Ui+, i.e., generates the same sequences {fit (v)}t=0,1,..., v Ui, as when applied only to Ui under zero initial data fi0 (v), v Ui. Therefore the subset {fit (v)}t=0,1,..., v Ui possesses all of the above-stated properties (a)-(f), including nite con vergence to a subset of values fi (v), v Ui, which is consistent with on Ui and is such that v Ui.

fit (u) N + 1, for u Ui+, u = ui. (30) Comparing (30) with (28), we conclude that the procedure (22) works over subset Ui+ independently of Ui, i.e., generates the same sequences 350 III. Теория принятия решений {fit (w)}t=0,1,..., w Ui+, as when applied only to Ui+ under a subset of initial data (26) for x Ui+. On the other hand, let us apply the procedure (22) to Ui+ under zero initial data: fi0 (w) = 0, w Ui+. Then because ui obviously is the unique minimal element in Ui+, we shall get for t = 1:

fi1 (ui ) = 0;

fi1 (u) = 1, for u Ui+, u = ui. (31) Comparing (31) with (29), we conclude that the procedure (22) on Ui+ under initial data set (26) generates values fit (w), t = 1, 2,..., which exceed exactly by N the corresponding values given under zero initial data. Latter values as it has been pointed out earlier must possess all Properties (a)-(f).

Hence the procedure (22) ensures nite convergence of the sequence subset {fit (w)}t=0,1,..., w Ui+, to the value subset fi (w), w Ui+, consistent with on Ui+ and such that fi (w) 2N, for all w Ui+.

N (32) Finally, it is easy to see that united set of values f i (u), u U = Ui Ui+, yields the function fi on U consistent with on the overall set U (due to (27), (28), and (32)).

Let us now construct a vectorial function f not only consistent with but representing the relation on U. Note that the above-constructed scalar function fi possesses the following property:

then fi (u) fi (ui ) If u = ui, and u ui, (33) (it follows from estimates (28) and (32)). Letting f = (f1,..., fN ), we get f consistent with on U such that (due to (33)), for every u, v U, we have the following additional property:

If u v, then f (u) f (v). (34) Condition (34) together with consistency of f with implies that f does represent on U. This ends the proof of Lemma 4 for the general vectorial case.

Remark. This constructive proof of the existence of a vectorial function f representing on U simultaneously gives an upper bound for its di mensionality: it is sucient to take f with no more than N = |U | scalar components.

B. Problems of extremization under binary relations. Let be a superiority relation on U (i.e., an asymmetrical binary relation). We call a problem of extremization under relation (in short, a -problem) the following problem of selection of the subset Y from a set X dened as Y = {y X| there is no x X such that x y}. (35) Conditions for universal reducibility Every -problem being considered as a mass problem (over all X X ) represents a special type of choice problem X Y, X X. The main question interesting us herein concerns the reducibility of a choice problem to the problem of extremization of a function (f -problem). It is convenient to reduce the solution of this question to the two steps: (1) applying a criterion of reducibility of a choice problem to a -problem, and (2) applying a criterion of reducibility of a -problem to an f -problem. Let us begin with the 1st step.

Denition 5. For an arbitrary choice problem X Y on X = 2U \{ }, we shall call the relation of pairwise-revealed superiority the binary relation P on U dened as uP v [u but not v is chosen from the pair {u, v}]. (36) for every u, v U.

Remark. Relation P in virtue of Denition 5 surely is asymmetrical, i.e., it is really a superiority relation.

Lemma 5. It a choice problem X Y on X = 2U \{ } is reducible to a problem of extremization under some superiority relation, then coincides with the pairwise-revealed superiority relation P for this problem.

Proof of the lemma is reduced to comparing the above denition (36) with the denition of choice (35) for the case X = {u, v}.

Lemma 5 implies directly the following criterion of reducibility of a choice problem to a -problem.

Lemma 6. For a choice problem X Y on X = 2U \{ } to be reducible to a -problem, it is necessary and sucient that for every X X the following is true:

Y = {y X| there is no x X such that xP y}. (37) Equation (37) will be called the Condorcet Principle. The essence of the Condorcet Principle is the requirement that a given choice problem X Y be reducible to the problem of extremization under its relation of pairwise-revealed superiority, i.e., to the -problem with = P. Lemma says that this requirement is not only sucient (which is trivial) but also necessary (which is declared by Lemma 5) for reducibility to a -problem.

Let us go to the 2nd step, to the statements concerning mutual redu cibility of -problems to f -problems.

Lemma 7. If a superiority relation on U is represented by a function f, then the problem of extremization under the relation and the problem of extremization of the function f on X = 2U \{ } are mutually reducible, one to another.

Proof of the Lemma is given by direct replacement of the relation x y by f (x) f (y) in the formulation of the -problem.

Lemma 7 opens the path of construction of direct counterparts of Lemmata 2 and 4 in terms of extremization problems:

Lemma 8. Each problem of extremization of a scalar (or vectorial) function f on U is reducible to the problem of extremization under a superiority relation which is a weak (resp., partial) order on U.

Lemma 9. Each problem of extremization under a superiority relation which is a weak (or partial) order on U is reducible to a problem of extremization of a scalar (resp., vectorial) function f on U.

Now we can synthesize the two steps of the reasoning and formulate the next lemma.

Lemma 10. For a choice problem X Y on X = 2U \{ } to be reducible to the problem of extremization of a scalar (or vectorial) function f, it is necessary and sucient that the following two conditions be fullled:

Proof. (a) Necessity: Let the given choice problem be reducible to the f problem with a scalar (or vectorial) function f. Then in virtue of Lemma 8, it is reducible also to the -problem with a relation being a weak (resp., partial) order. The latter in virtue of Lemma 6 implies satisfying the Condorcet Principle for the initial problem (condition(1)). Moreover, in virtue of Lemma 5, its relation P must coincide with the relation, and must be a weak (resp., partial) order (condition (2)).

hence P (b) Suciency: Let the given choice problem satisfy conditions (1) and (2) of the lemma. Then due to condition (1), in virtue of Lemma 6 the given problem is reducible to the problem of extremization under the relation = P, which is by condition (2) a weak (or partial) order. But then in virtue of Lemma 9 the initial problem is reducible also to the problem of extremization of a scalar (resp., vectorial) function f.

Remark. Now a proof of Lemma 1 can be obtained as a simple corollary from Lemma 10. Really, in virtue of Lemma 10 for reducibility of a vectorial -problem to a scalar problem, it is necessary and sucient that the superiority relation P having form xP y (x) (y) is a weak order.

It is equivalent to transitivity of the respective indierence relation I of Conditions for universal reducibility the form xIy (x) (y) which is just equivalent to the absence of -triads in U.

C. Proof of Theorem 2. Apply the routine of two-step analysis given in the form of the two conditions in Lemma 10 to the analysis of reducibility of a vectorial-vectorial, -problem to some one-stage vectorial or scalar problem.

(1) Conditions of satisfying the Condorcet Principle for the, -prob lem. To begin with let us write the pairwise-revealed superiority relation P for the, -problem (3):

uP v [(u) (v) or ((u) (v) and (u) (v))]. (38) The testing of the Condorcet Principle (37) is reduced to the testing of the two statements which together form the Principle. These two statements (for any X X and for the choice Y from X), are, respectively:

If y X and if there is no x X such that xP y, then y Y (39) (Direct Condorcet Condition), and:

If y Y, then there is no x X such that xP y (40) (Converse Condorcet Condition).

Let us rst examine the Direct Condorcet Condition (39). Assume that in a, -problem y Y holds for some X X and for some y X. The form of two-stage problem (3)–(4) shows that it is possible in one of two cases: (a) there exists z X such that (z) (y), and then y V ;

or (b) y V, but there exists z X such that z V and (z ) (y). In the case (b) there necessarily holds (z ) (y). It is easy to see that in the case (a), we have z P y and in the case (b), we have z P y. Therefore, if y holds for no x X, then necessarily y Y.

Consider now the Converse Condorcet Condition (40). Assume that the given, -problem violates this condition, viz, there exist X X and x, y X such that y Y but xP y. From y Y and x X follows (x) (y). (41) In virtue of xP y then by (41) (x) (y) and (x) (y). (42) Assumption y Y implies x V by (42). Therefore there exists a third element z X such that (z) (x). (43) 354 III. Теория принятия решений On the other hand, in virtue of y Y, z X, we have (z) (y). (44) Furthermore, the relation (y) (z) is also impossible because in this case in virtue of (25) and transitivity of, we would obtain (y) (x), contrary to (42). Hence (y) (z), which being combined with (44) yields (y) (z). (45) Now if for the given z there exists one more element z X such that (z ) (z), then it together with (43) yields (z ) (x). If, further more, there exists z X such that (z ) (z ), then in a similar way (z ) (x), etc. Due to the niteness of X and the acyclicity of, there always exists an element in X, we preserve the notation z for it, such that (43) holds and there is no s X such that (s) (z), and hence z V. But then for y Y V, we obtain (z) (y). (46) Relations (42), (43), (44), and (46) together imply that the triple x, y, z is a, -1-triad (letting u = x, v = z, w = y in the denition (8), (18.1)).

Conversely, if there exists a, -1-triad u, v, w in U, then letting X = {u, v, w}, we obtain w Y, v X, but v P w, contrary to the Converse Condorcet Condition. Therefore the Converse Condorcet Condition is not fullled if and only if there exists a, -1-triad in U.

So we have proved the following Lemma.

Lemma 11. For a vectorial-vectorial, -problem to satisfy the Condor cet Principle, it is necessary and sucient that, -1-triads were absent in U.

(2) Conditions of order for relation P in the, -problem. Now we shall nd conditions for transitivity of a superiority relation = P and also of a corresponding indierence relation which will be denoted here by I.

Lemma 12. Let a vectorial-vectorial, -problem satisfy the Condorcet Principle, and let P be the pairwise-revealed superiority relation for this the corresponding indierence relation. Then:

problem and I (a) for transitivity of P, it is necessary and sucient that, -2 triads were absent in U ;

(b) for transitivity of I, it is necessary and sucient that, -3 and, -3-triads were absent in U.

Proof of lemma. (a) Nontransitivity of P would imply that there is a triple x, y, z of elements in U, such that xP y, y P z, but not xP z. (47) Conditions for universal reducibility There are three possible systems of relations between values and on the triple x, y, z which according to (38) give (47):

6. Conditions for the elimination of one stage in the two-stage problem Pose the question of the reducibility of a two-stage, generally vectorial vectorial,, -problem not only to some one-stage problem of extremiza tion of some function f, but more denitely, to the problem of extremiza tion of just the same function f = or f = which is present in one of two stages of the initial problem. Speaking in a dierent way, the question is:

under what conditions is it possible to discard, i.e., simply to “throw out,” one of the stages in the two-stage problem without changing its solution set?

It is clear that these conditions must be surely more strict than the gen eral condition of the reducibility of a two-stage, -problem to a one-stage problem of extremization of some arbitrary function f. This latter condi tion has been obtained in Theorem 2 for the general case of the vectorial vectorial, -problem in the form of prohibition of special three-element structural formations in U, viz,, -1- and, -2-inconsistent triads. It turns out that the sought-after more strict conditions of reducibility can be formulated in the form of stronger prohibitions. Namely, we must prohibit certain two-element structural formations, “dyads”, implicitly contained in the above-mentioned triads.

Observe that in this Lemma and in the sequel we consider, -incon sistent but, -noncoherent (with reverse order of and !) dyads.

Theorem 3. For a two-stage vectorial-vectorial, -problem to be re ducible to its rst (or second) stage, i.e., to the problem of extremization of the vectorial function (respectively, ), it is necessary and sucient that in U any, -noncoherent dyads (respectively,, -inconsistent dyads) be absent.

Before proving the Theorem, we note that the condition of absence of, -inconsistent dyads in U can be obviously presented in the following equivalent form:

(x) (y) (x) (y), for all x, y U (56) (a condition of, -consistency). Similarly, let us consider the condition of absence of, -noncoherent dyads, i.e., of pairs u, v U such that (u) (v) but (u) (v). It is easy to see that this condition can be presented in the following equivalent form: (u) (v) (u) (v), for all u, v U, which is also equivalent to (x) (y) (x) (y), for all x, y U (57) (condition of, -coherence). Hence for proving Theorem 3, it is sucient to prove the following equivalent statement:

For the reducibility of a given, -problem to the -problem or to the -problem, it is necessary and sucient to meet the, -coherence condition (57) or, respectively, the, -consistency condition (56).

Remark. Note that the, -consistency condition (56) coincides with the requirement of consistency (in the sense of Def. 4) of the function with the relation represented by the function.

Now consider two versions of superiority relations represented by functions and :

u v (u) (v) (58) and v (u) (v), u (59) respectively. The consideration of the expressions (58) and (59) together with the expression (38) dening the pairwise-revealed superiority relation P in the, -problem yields directly the following lemma.

358 III. Теория принятия решений Lemma 14. For the relation P from (38) to coincide on U with the relation from (58) or from (59), it is necessary and sucient to meet the, -coherence condition (57) or, respectively, the, -consistency condition (56).

Proof of Theorem 3. Assume that a given, problem is reduced to the -problem (or to the -problem) with the function (or ) just the same as is present in the initial, -problem. But each -problem (resp., -problem) in its own turn is reducible (due to Lemma 7) to the problem of extremization under the relation, represented by the function (resp., ), i.e., dened from (58) (resp., (59)). Therefore, the initial, -problem must be reducible to the problem of extremization under the relation from (58) (resp., (59)). But then due to Lemma 5 the relation must coincide with the relation P for the initial, -problem. Hence due to Lemma 14 the condition (57) (resp., (56)) necessarily is satised.

Conversely, let the condition (57) or (56) be satised, i.e., equivalently, the condition of absence of, -noncoherent or, -inconsistent dyads in U be fullled. Then due to Lemma 13 in every case, -1- and, -2 inconsistent triads must be absent in U. Hence by Theorem 2 the initial, -problem must be reducible to a problem of extremization of some (generally vectorial) function f. Therefore, according to Lemma 10, the, -problem must satisfy the Condorcet Principle, i.e., it must be re ducible to the problem of extremization under the relation P of the form (38). But in virtue of condition (57) (or, respectively, (56)) by Lemma the relation P coincides with the relation from (58) (resp., from (59)), i.e., with the relation represented by the function (resp., by ). Hence in virtue of Lemma 7 the given, -problem will be reducible to the problem or to the -problem, respectively.

The Theorem is proved.

7. An illustrative example Consider a simple mechanism for generating a two-stage vectorial-scalar extremization. Let an n-dimensional criterial function (x) = (1 (x),..., n (x)) on a set U be given, and let a xed “ideal point” in the n dimensional space of vectors of criterial values also be given. Suppose that a choice of elements from an admissible set X X is implemented in two stages. At the rst stage the set V of Pareto-optimal elements of X by is selected. The set V corresponds to the Pareto subset (V ) of the admissible set (X) in -space. At the second stage those elements y are selected from V which are represented in the -space by the points (y) nearest to the ideal point. Hence the whole problem consists of two stages:

Conditions for universal reducibility (1) extremization (viz, maximization) of over X, i.e., selection of the set V = Arg maxxX (x), and (2) extremization (viz, minimization) of the distance d from the points of the set (V ) to, i.e., maximization over V of the function (v) = d((v), ), or speaking dierently, selection of the set Y = Arg minvV d((v), ) = Arg maxvV (v).

Consider the two cases.

Case (a) The ideal point is not exceeded on the set U :

, i : i i where = max i (x).

i xU It is easy to see in this case that condition (56) of, -consistency (absence of, -inconsistent dyads) is satised. Hence due to Theorem 3 the given two-stage problem is reducible to its second stage, i.e., to nding elements x in X minimizing the distance d((x), ) from the point (x) to the ideal point. Preliminary selection of the Pareto set by the vectorial criterion in this case is redundant from the theoretical point of view.

Case (b). The ideal point is exceeded on U at least in one coordi nate i:

i : i.

i In this case some, -inconsistent dyad, or moreover,, -1-inconsis tent triad can be formed in U. An example of such a situation is presented in Fig. 1. In this particular example the function (x) is two-dimensional, the scalar function (x) is dened as the Euclidean distance, with the op posite sign, on the plane between points (x) and. For the conguration of points in the -space in Fig. 1, we have d((x), ) d((y), ) d((z), ), and hence (x) (y) (z), which together with vectorial relations (x) (y), (y) (z), (x) (z) makes the triple x, y, z a, -1-triad (with u = x, v = z, w = y in denitions (8), (18.1)). Therefore, in virtue of Theorem 2 such a two-stage, -problem not only is irreducible to its second stage, but moreover, it is not reducible to any one-stage extremization problem at all.

We can demonstrate this directly. It is evident that Pareto-optimal elements in the set X = {x, y, z} are y and z but not x. In the pair y and z 360 III. Теория принятия решений in -space, the nearest to the ideal w turns out to be the element y which will be the only eventually chosen element from X. At the same time in a narrower set X = {x, y} X both elements x, y will be Pareto-optimal, and the nearest to of these two will be not y but x which will be chosen nally in spite of the presence of y in X.

2 T $ e $$ $$$ d (y) $ d(z) d (x) E Fig. If the given two-stage problem were reducible universally to some one stage problem of extremization of some, even vectorial, function f, then the above solutions of the initial problem would have yielded the following.

Due to the choice of y from X = {x, y, z}, we would have f (y) f (x), but due to the choice of x from X = {x, y}, f (x) f (y). This is impossible if f does not depend on X.

So the consideration of both cases, (a) and (b), shows us that a two stage problem of the “selection of the nearest to the ideal” points in the Pareto set can demonstrate dierent properties. Depending on mutual disposition of the initial set and the ideal point, the problem can reveal both universal reducibility, and irreducibility to one-stage extremization.

Remark. In the considered example of a two-stage problem the ideal point is assumed to be xed, independent of X. Sometimes one considers a little dierent statement of such a problem when is a function of X which may be implicit. So, e.g., if we let i : i = maxxX i (x), then = (1,..., n ) obviously depends on X : = (X). In this case the initial two-stage problem is not a, -problem of the form (3). Indeed, its second stage, maximization of the function = d((x), (X)), is not an extremization problem of the form (1), because here the criterial function does depend on X as on a parameter: = X (x). Hence a problem with a unxed ideal point does not fall under the above scheme of analysis at all.

This remark once again reminds us that the abstract form of the extremization problem (1) adopted here, though very general, nevertheless imposes an essential requirement. This requirement lies in the criterial function independence of set X being variated in the “mass” statement Характеризация иерархии уровней полезности of the problem. This requirement, pointed out in the Introduction (see Remark in Sect. 1), is just one in which underlies the exact statement of the question concerning the universal reducibility of a problem to one-stage extremization and makes the question nontrivially solvable in principle.